Um, that's not what he said. He said that he is England to his brothers, but the UK to everyone else. This must mean that he is the anthropomorphism of England, and not anthropomorphism of the UK/Britain. As long as Scotland and Wales are proven to exist, he cannot be the anthropomorphism of the UK/Britain. All he does is represent them all at meetings/to other nations. Scotland, Wales and Northern Ireland are anthropomorphisms in their own right.

It is also not possible because there are times in history where the UK did not exist.

The reason he represents them is because that's how our political system works, like you said. But his proper name is not 'UK' or 'Britain', it has to be 'England' because that is what he the anthropomorphism of.